On the product formula on non-compact Grassmannians

Abstract : We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star \delta_{e^Y}^\natural$ of two orbital measures on the symmetric space ${\bf SO}_0(p,q)/{\bf SO}(p)\times{\bf SO}(q)$, $q>p$. We prove sharp conditions on $X$, $Y\in\a$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for $\SO_0(p,q)/\SO(p)\times\SO(q)$ will also serve for the spaces ${\bf SU}(p,q)/{\bf S}({\bf U}(p)\times{\bf U}(q))$ and ${\bf Sp}(p,q)/{\bf Sp}(p)\times{\bf Sp}(q)$, $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $\delta_{e^X}^\natural$.
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Submitted on : Friday, November 30, 2012 - 12:12:53 PM
Last modification on : Wednesday, December 19, 2018 - 2:08:04 PM
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  • HAL Id : hal-00759238, version 1
  • ARXIV : 1212.0002

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Piotr Graczyk, Patrice Sawyer. On the product formula on non-compact Grassmannians. 2012. ⟨hal-00759238⟩

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